Approximating Eigenvalues via the Landscape function
- Shiwen Zhang, UMass Lowell
- September 14, 12:30-1:30 p.m., In-person and Virtual Meeting
Abstract: In this talk, we study the approximation of eigenvalues via the Landscape function for some random Schrodinger operators as well as some random band matrices. We first give a brief review of the localization landscape theory. Then we focus on some one-dimensional model and show that ratio the ground state energy and the minimum of the landscape potential approaches pi^2/8 as the size of the system approaches infinity. We then present numerical stimulations for the same asymptotic behavior for excited states of some random band matrices. Finally, we discuss conjectures and open problems based on these numerical results and their relation to random graphs as well as higher dimensional models.
Van der Waerden’s theorem is (nearly) 100: the Ramsey heuristic and the Algebraic Structure of the Stone–Čech Compactification
- John Johnson, Ohio State University
- September 28, 12:30-1:30 p.m., Virtual Meeting
An observation of Kakeya and Morimoto states van der Waerden’s theorem is equivalent to the assertion that “sets of positive integers with bounded gaps (a notion of size) contains arbitrarily long arithmetic progressions.” This observation and related techniques are prototypical of a ubiquitous heuristic: notions of size and their structures imply some interesting (combinatorial) pattern. Motivated by this classical result and its modern generalizations, I’ll illustrate via a suggestive visualization and show how precise instances of this heuristic translate into interesting questions and applications of the algebraic structure of the Stone–Čech compactification. I’ll highlight those generalizations that, so far, seem to require a combination of algebraic and combinatorial techniques.
(Based on previous work with Vitaly Bergelson and Joel Moriera; Cory Christopherson; and Florian Richter.)
A Simple and Accurate Numerical Method for Singular and Near-singular Integration
- Bobbie Wu, UMass Lowell
- October 12, 12:30-1:30 p.m., In-person and Virtual Meeting
Boundary Value Problems (BVPs) are ubiquitous in engineering and scientific applications. One of the most robust and accurate methods for solving BVPs is the Boundary Integral Equation Method, which has the great advantage of dimensionality reduction: all of the unknowns reside on the boundary surface instead of in the volume. A key challenge when solving integral equations is that special quadrature methods are required to discretize the underlying singular and near-singular integral operators. Accurate discretization of these operators is especially important in, for example, problems that involve close structure-structure or fluid-structure interactions. In this talk, we present some recent advancements on singular and near-singular numerical integration based on the Trapezoidal rule -- one of the simplest quadrature methods.
A Neural Network Approach for Homogenization of Multiscale Problems
- Jihun Han, Dartmouth College
- October 19, 12:30-1:30 p.m., In-person and Virtual Meeting
Abstract: We propose a neural network-based approach to the homogenization of multiscale problems. The proposed method uses a derivative-free formulation of a training loss, which incorporates Brownian walkers to find the macroscopic description of a multiscale PDE solution. Compared with other network-based approaches for multiscale problems, the proposed method is free from the design of hand-crafted neural network architecture and the cell problem to calculate the homogenization coefficient. The exploration neighborhood of the Brownian walkers affects the overall learning trajectory. We determine the bounds of micro- and macro-time steps that capture the local heterogeneous and global homogeneous solution behaviors, respectively, through a neural network. The bounds imply that the computational cost of the proposed method is independent of the microscale periodic structure for the standard periodic problems. We validate the efficiency and robustness of the proposed method through a suite of linear and nonlinear multiscale problems with periodic and random field coefficients. This is joint work with Yoonsang Lee.
Self-similar Blowup Phenomena in Nonlinear Evolution Equations
- Irfan Glogic, University of Vienna
- October 26, 12:30-1:30 p.m., Virtual Meeting
Riemann’s Non-differentiable Function: A Turbulent History
- Daniel Eceizabarrena, UMass Amherst
- November 9, 12:30-1:30 p.m., In-person and Virtual Meeting
Abstract: Around 1860, Riemann challenged the beliefs of the time that a continuous function must have a derivative. The function he devised was not given by the typical closed expression, but rather by an infinite sum which he claimed to be continuous everywhere but to have a derivative nowhere. Claim is the correct word, for he did not prove it, leaving with it two long-lasting footprints: a big problem, unsolved until 1970, and a revolution and the consequent establishments of the foundations of modern mathematics. This function came to be known as Riemann's non-differentiable function which, turns out, is almost nowhere differentiable. Recently, this rich analytic structure has proved itself valuable in the setting of turbulence, one of the biggest open problems in mathematical physics. I will begin the talk with a broad historic overview to then describe the structure of the function and the results that were progressively proved for it. After that, I will aim at the role that it plays in turbulence, introducing elements like vortex filaments, multifractality and intermittency, together with the relevant mathematical tools used in their study.
Improving Numerical Accuracy for the Viscous-Plastic Formulation of Sea Ice
- Tongtong Li, Dartmouth College
- November 16, 12:30-1:30 p.m., In-person and Virtual Meeting
Abstract: Accurate modeling of sea ice dynamics is critical for predicting environmental variables, which in turn is important in applications such as navigating ice breaker ships, and has led to extensive research in both modeling and simulating sea ice dynamics. The most widely accepted model is the one based on the viscous-plastic formulation introduced by Hibler, which is intrinsically difficult to solve numerically due to highly nonlinear features. In particular, sea ice simulations often significantly differ from satellite observations. In this study we focus on improving the numerical accuracy of the viscous-plastic sea ice model. We explore the convergence properties for various numerical solutions of the sea ice model and in particular examine the poor convergence seen in existing numerical methods. To address these issues, we demonstrate that using higher order methods for solving conservation laws, such as the weighted essentially non-oscillatory (WENO) schemes, is critical for numerically solving viscous-plastic formulations whenever the solution is not smooth. Moreover, WENO yields higher order convergence for smooth solutions than standard central differencing does. Our numerical examples verify this, and in particular by using WENO, we are able to resolve the discontinuities in the sharp features of sea ice covers. We also propose an approach utilizing the idea of phase field method to develop a potential function method which naturally incorporates the physical restrictions of ice thickness and ice concentration in transport equations. Our approach results in modified transport equations with extra forcing terms coming from potential energy function, and has the advantage of not requiring any post-processing procedure that might introduce discontinuities and thus ruin the solution behavior.
Old and New Questions in the Regularity Theory of the d-bar-Neumann Problem
- Gian Maria Dall'Ara, Scuola Normale Superiore Pisa
- November 30, 12:30-1:30 p.m., Virtual Meeting
Abstract: In this talk, I will discuss various questions in the analysis of the d-bar-Neumann problem, a classical noncoercive boundary value problem for the Laplacian whose regularity theory is of great importance in complex analysis and geometry. Despite the deep results established in the last 60 years (starting with seminal work by D. Spencer, J. J. Kohn, J. Nirenberg, E. Stein, C. Fefferman,...), several central questions remain unanswered. I will try to survey classical and more recent approaches to the problem, highlighting the variety of techniques available, ranging from algebraic geometry to mathematical physics.